Nicolai V. Krylov

Existence of strong solutions for Itô's stochastic equations via approximations

SummaryGiven strong uniqueness for an Itôs stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on “any” probability space by using, for example, Eulers polygonal approximations. Stochastic equations in ℝd and in domains in ℝd are considered.

ON REGULARITY OF TRANSITION PROBABILITIES AND INVARIANT MEASURES OF SINGULAR DIFFUSIONS UNDER MINIMAL CONDITIONS

Let A = (aij ) be a matrix-valued Borel mapping on a domain Ω ⊂ R d , let b = (bi ) be a vector field on Ω, and let LA, b ϕ = a ij ∂ x i ∂ xj ϕ + bi ∂ xi ϕ. We study Borel measures μ on Ω that satisfy the elliptic equation LA, b *μ = 0 in the weak sense: ∫ LA, b ϕ dμ = 0 for all ϕ ∈ C 0 ∞ (Ω). We prove that, under mild conditions, μ has a density. If A is locally uniformly nondegenerate, A ∈ H loc p, 1 and b ∈ L loc p for some p > d, then this density belongs to H loc p, 1. Actually, we prove Sobolev regularity for solutions of certain generalized nonlinear elliptic inequalities. Analogous results are obtained in the parabolic case. These results are applied to transition probabilities and invariant measures of diffusion processes.

Parabolic and elliptic equations with VMO coefficients

An Lp-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class VMOx, which, in particular, contains all functions independent of x. Weak uniqueness of the martingale problem associated with such equations is obtained.

Elliptic Differential Equations with Coefficients Measurable with Respect to One Variable and VMO with Respect to the Others

We prove the unique solvability of second order elliptic equations in nondivergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the Martingale problem associated with the elliptic equations.

On L p -theory of stochastic partial differential equations in the whole space

It is shown that equations like \[ du = \left( {a^{ij} u_{x^i x^j } + b^i u_{x^i } + cu + f} \right)dt + \left( {\sigma ^{ik} u_{x^i } + \nu ^k u + g^k } \right)dw_t^k ,\quad t > 0, \] with variabl...

AW 2 n -theory of the Dirichlet problem for SPDEs in general smooth domains

SummaryStochastic partial differential equations in smooth domains are considered in functional spaces of Sobolev type. The spaces are defined with the help of certain weights, which allow the derivatives of functions from these spaces to blow up near the boundary. Existence and uniqueness theorems are obtained.

Elliptic and Parabolic Second-Order PDEs with Growing Coefficients

We consider a second-order parabolic equation in ℝ d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the L ∞-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.

Stochastic partial differential equations with unbounded coefficients and applications. III

We extend the results of the earliei papers in this series [Stochastics and Stochastics Reports 27, 129-150 and 27, 189-233] on the approximation of SPDEs to the case of degenerate systems of SPDEs with unbounded coefficients. Some examples and applications are also considered

On the rate of convergence of finite-difference approximations for Bellman equations with constant coefficients

We consider parabolic Bellman equations with Lipschitz coefficients. Error bounds of order h1/2 for certain types of finite-difference schemes are obtained.

On averaging principle for diffusion processes with null-recurrent fast component

An averaging principle is proved for diffusion processes of type (X[var epsilon](t),Y[var epsilon](t)) with null-recurrent fast component X[var epsilon](t). In contrast with positive recurrent setting, the slow component Y[var epsilon](t) alone cannot be approximated by diffusion processes. However, one can approximate the pair (X[var epsilon](t),Y[var epsilon](t)) by a Markov diffusion with coefficients averaged in some sense.